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Theorem cantnfres 9435
Description: The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
cantnfrescl.d (𝜑𝐷 ∈ On)
cantnfrescl.b (𝜑𝐵𝐷)
cantnfrescl.x ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋 = ∅)
cantnfrescl.a (𝜑 → ∅ ∈ 𝐴)
cantnfrescl.t 𝑇 = dom (𝐴 CNF 𝐷)
cantnfres.m (𝜑 → (𝑛𝐵𝑋) ∈ 𝑆)
Assertion
Ref Expression
cantnfres (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)))
Distinct variable groups:   𝐵,𝑛   𝐷,𝑛   𝐴,𝑛   𝜑,𝑛
Allowed substitution hints:   𝑆(𝑛)   𝑇(𝑛)   𝑋(𝑛)

Proof of Theorem cantnfres
Dummy variables 𝑘 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfrescl.d . . . . . . . . . . . . 13 (𝜑𝐷 ∈ On)
2 cantnfrescl.b . . . . . . . . . . . . 13 (𝜑𝐵𝐷)
3 cantnfrescl.x . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋 = ∅)
41, 2, 3extmptsuppeq 8004 . . . . . . . . . . . 12 (𝜑 → ((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅))
5 oieq2 9272 . . . . . . . . . . . 12 (((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅) → OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐷𝑋) supp ∅)))
64, 5syl 17 . . . . . . . . . . 11 (𝜑 → OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐷𝑋) supp ∅)))
76fveq1d 6776 . . . . . . . . . 10 (𝜑 → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))
873ad2ant1 1132 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))
98oveq2d 7291 . . . . . . . 8 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = (𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
10 suppssdm 7993 . . . . . . . . . . . . 13 ((𝑛𝐵𝑋) supp ∅) ⊆ dom (𝑛𝐵𝑋)
11 eqid 2738 . . . . . . . . . . . . . . 15 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
1211dmmptss 6144 . . . . . . . . . . . . . 14 dom (𝑛𝐵𝑋) ⊆ 𝐵
1312a1i 11 . . . . . . . . . . . . 13 (𝜑 → dom (𝑛𝐵𝑋) ⊆ 𝐵)
1410, 13sstrid 3932 . . . . . . . . . . . 12 (𝜑 → ((𝑛𝐵𝑋) supp ∅) ⊆ 𝐵)
15143ad2ant1 1132 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐵𝑋) supp ∅) ⊆ 𝐵)
16 eqid 2738 . . . . . . . . . . . . . 14 OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐵𝑋) supp ∅))
1716oif 9289 . . . . . . . . . . . . 13 OrdIso( E , ((𝑛𝐵𝑋) supp ∅)):dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅))⟶((𝑛𝐵𝑋) supp ∅)
1817ffvelrni 6960 . . . . . . . . . . . 12 (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ ((𝑛𝐵𝑋) supp ∅))
19183ad2ant2 1133 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ ((𝑛𝐵𝑋) supp ∅))
2015, 19sseldd 3922 . . . . . . . . . 10 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ 𝐵)
2120fvresd 6794 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛𝐷𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)))
2223ad2ant1 1132 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → 𝐵𝐷)
2322resmptd 5948 . . . . . . . . . 10 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐷𝑋) ↾ 𝐵) = (𝑛𝐵𝑋))
2423fveq1d 6776 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛𝐷𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)))
258fveq2d 6778 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
2621, 24, 253eqtr3d 2786 . . . . . . . 8 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
279, 26oveq12d 7293 . . . . . . 7 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) = ((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))))
2827oveq1d 7290 . . . . . 6 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧) = (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧))
2928mpoeq3dva 7352 . . . . 5 (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)))
306dmeqd 5814 . . . . . 6 (𝜑 → dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)))
31 eqid 2738 . . . . . 6 On = On
32 mpoeq12 7348 . . . . . 6 ((dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)) ∧ On = On) → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)))
3330, 31, 32sylancl 586 . . . . 5 (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)))
3429, 33eqtrd 2778 . . . 4 (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)))
35 eqid 2738 . . . 4 ∅ = ∅
36 seqomeq12 8285 . . . 4 (((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)) ∧ ∅ = ∅) → seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅))
3734, 35, 36sylancl 586 . . 3 (𝜑 → seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅))
3837, 30fveq12d 6781 . 2 (𝜑 → (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅))) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅))))
39 cantnfs.s . . 3 𝑆 = dom (𝐴 CNF 𝐵)
40 cantnfs.a . . 3 (𝜑𝐴 ∈ On)
41 cantnfs.b . . 3 (𝜑𝐵 ∈ On)
42 cantnfres.m . . 3 (𝜑 → (𝑛𝐵𝑋) ∈ 𝑆)
43 eqid 2738 . . 3 seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)
4439, 40, 41, 16, 42, 43cantnfval2 9427 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅))))
45 cantnfrescl.t . . 3 𝑇 = dom (𝐴 CNF 𝐷)
46 eqid 2738 . . 3 OrdIso( E , ((𝑛𝐷𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐷𝑋) supp ∅))
47 cantnfrescl.a . . . . 5 (𝜑 → ∅ ∈ 𝐴)
4839, 40, 41, 1, 2, 3, 47, 45cantnfrescl 9434 . . . 4 (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ (𝑛𝐷𝑋) ∈ 𝑇))
4942, 48mpbid 231 . . 3 (𝜑 → (𝑛𝐷𝑋) ∈ 𝑇)
50 eqid 2738 . . 3 seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)
5145, 40, 1, 46, 49, 50cantnfval2 9427 . 2 (𝜑 → ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅))))
5238, 44, 513eqtr4d 2788 1 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  Vcvv 3432  cdif 3884  wss 3887  c0 4256  cmpt 5157   E cep 5494  dom cdm 5589  cres 5591  Oncon0 6266  cfv 6433  (class class class)co 7275  cmpo 7277   supp csupp 7977  seqωcseqom 8278   +o coa 8294   ·o comu 8295  o coe 8296  OrdIsocoi 9268   CNF ccnf 9419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-seqom 8279  df-1o 8297  df-oadd 8301  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-oi 9269  df-cnf 9420
This theorem is referenced by: (None)
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